Shareholder rights law firm Johnson Fistel, LLP ( is investigating whether Amplitude, Inc. (NASDAQ:AMPL), any of its executive officers or others

AMPL SHAREHOLDER ALERT: Johnson Fistel Investigates Amplitude

Despite recent volatility, U.S. equities markets have had a much better year than expected in 2023. The S&P 500 and Nasdaq are only returning 9.8% and 25.5% since the start of the year. Geopolitical events that could move the world in different directions and soaring treasury yields were to blame for the market sell-off last week.

3 Struggling Stocks to Sell in November Before They Implode

InvestorPlace - Stock Market News, Stock Advice & Trading Tips Despite recent volatility, U.S. equities markets have had a much better year

These WWE tag teams have crafted an unforgettable legacy in the promotion, supported by some truly amazing finishers.

10 Best WWE Tag Teams Ever, Ranked By Their Finisher

Figure 3. Silicon Creations DP RX JTOL vs DP mask DP Main Link Source Device Electrical Requirements: The DP Source device must consist of a transmitter that can transmit the required data rate at specified amplitude levels. The TX must also support ...

High Speed, Low Power and Flexibility Drive DisplayPort's Increasing Popularity

To determine the effect of melanin content on conventional amplitude-based PA images, as well as the impact of SLSC beamforming, they calculated the signal-to-noise ratio (SNR) of the radial artery running through the forearm and the level of surrounding ...

High-quality diagnostic imaging – for all skin tones

The Endless universe sprang to life from the strategic wranglings of Endless Space, a 4X turn-based strategy game that frankly couldn’t be further from Endless Dungeon if you tried. However, this shared sci-fi system plays host to an array of different races and technologies, many of which you’ll encounter in the deep levels of the ship.

Endless Dungeon Review

Endless Dungeon, the latest nugget from development studio Amplitude Studios, is shaping up to be a must-have for video game lovers! Published by publisher SEGA, the game awaits you on October 19, 2023 on PC, PS4, PS5, Xbox One, Xbox Series and Nintendo Switch, marking a new stage in the evolution of tactical roguelikes.

Endless Dungeon: our review of Amplitude Studios' new hit!

Pomerantz LLP is investigating claims on behalf of investors of Amplitude, Inc. ("Amplitude" or the "Company") (NASDAQ:AMPL). Such investors are advised to contact Robert

SHAREHOLDER ALERT: Pomerantz Law Firm Investigates Claims On Behalf of Investors of Amplitude, Inc. - AMPL

LIGO first became world-famous in Sept. 2015, when it detected gravitational waves from merging black holes for the first time. These ripples had traveled for around 1.4 billion years, squashing and squeezing spacetime as they made their way through the universe .

LIGO gravitational wave detector breaks 'quantum limit' to find deep universe black hole collisions

Amplitude

amplitude

It’s 10:15pm. The Amplify conference starts the next morning, and I just found out that I would be attending. I could not wait. I received a pass to Amplify because of the generosity of the Advancing Women in Product (AWIP) community — AWIP is a organization whose goal is to empower the next generation of women product leaders. So there I was on the morning of October 9th, nibbling on my mini breakfast croissant, bumping against the shoulders of product leaders, anticipating the start of a jam-packed, learning-rich, & Ritual Roasters coffee-filled day. If I think about the learnings I drew out of the conference, I’d say they weren’t necessarily profound nuggets of wisdom — they’re more like gentle reminders. Things I already know but tend to forget when building products. Here’s what I learned: Growth is more than a funnel. It’s a lot of different loops. I often break down products via the Pirate Funnel (Awareness, Acquisition, Activation, Retention, Revenue or AAARR for short). But Casey Winters, (VP Product at Eventbrite, formerly growth at Pinterest & Grubhub, among others), argues that that framework prioritizes feeding only into the top of the funnel. When in fact, our products are ongoing feedback mechanisms, aka Loops. Casey Winters interviewing Elena Verna Macro loops stitch our micro loops over time. We have 3 general types of loops: Viral loops are when users organically share or are incentivized to refer other friends back to your product. Content loops are classified based on the of content generated. These can be company (HubSpot, WebMD), user (Pinterest) or supply (Grubhub) generated. type Paid loops vary based on who does the work to distribute the product. These can be ads, sales teams or integrations via partnerships. These growth loops paint a better picture than a funnel, and when architected like this you’ll reap the best outcomes. We can make a difference when we look at products from the lens of social science. I love Dan Ariely (best-selling author of Predictably Irrational, behavioral economics professor at Duke University). I love behavioral economics. I love Dan Ariely’s talk on behavioral economics for products. Dan Ariely giving the keynote! Dan performed a research study in Kenya with the goal of finding ways to help Kenyans save money. He tried several tactics, with mediocre results: injecting friction to make it more difficult to take out money, sending guilt-filled reminders to save, and introducing loss aversion in saving. But the tactic that worked the best by far was so simple and effective: if they saved money that day then they could scratch that day off on a calendar using a coin. The coin method made them aware of their daily actions of saving, and also injected social accountability from their family members. A simple, yet effective change in habit that made the often invisible act of saving money, extremely visible everyday. Another great talk on behavior was from Tatyana Mamut. Her talk offered tactical advice in dealing with bad Internet behavior such as malicious accounts and fake news. As a company scales, entire departments are dedicated to taking down these bad actors before they infect the rest of the user base. These examples were simple yet powerful examples telling us, “hey, our users are humans, too.” We’ve got a lot to learn from a behavioral scientist’s lens. Data’s important, but use it in context. Quantitative data’s so hot these days. And why not? It gives great insights into WHAT our users are doing. But the WHY is also equally important — and we need to marry both quantitative and qualitative data. When initially measuring the success of a product, qualitative data might weigh more heavily. Sarah Tavel from Bessemer ventures emphasizes drawing from our First Principles and front loading user research. Our empathy skills in asking the right questions & observing raw human behavior comes first, until we can really couple the user research with data at scale. “Product managers are the owners of a unique definition of a problem worth solving.” I this definition of a PM from Eric Vishria (General Partner at Benchmark Capital). Let’s break that down. love This definition is great. We can read it and say “oh, yeah, of course.” But we forget to live its principle. We forget to take ownership when things don’t go as planned. We forget to clearly define our goals & problems for everyone else to act on. We forget we are not the solution creators but the conduit to unlock everyone else’s creative potentials. If we can remember this holistic definition, we become better product managers. Some friends from the conference The product leader can understand the systematic complexities of the products they build. I like the analogy that the partners at Benchmark Capital highlighted: We should not lose the forest for the trees. It’s easy to silo into that one northstar metric, or be hyperfocused in building out that one feature. But the best product leaders can look at the forest the complexities of what they’re building. The PM needs to ask themself, PMs need to understand the wider, not often thought of implications of the products they put out into the world. and Does it affect other product metrics? Will this negatively impact specific cultures? What are the ethical risks associated with this design change? As a bonus, my 6th takeaway is that Ritual Coffee Roasters is . (I had one too many cups.) the best Thank you, Advancing Women in Product for the opportunity to learn from all these amazing product minds. I’m already on the waitlist for next years Amplify conference. Congratulations to Amplitude for a successful conference!

Amplify Conference 2018

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar B Derivation of Marginal Likelihood with Stochastic Field Amplitude The full signal in time space is given by Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood: where This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar B Derivation of Marginal Likelihood with Stochastic Field Amplitude The full signal in time space is given by Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood: where This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Derivation of Marginal Likelihood with Stochastic Field Amplitude

What is the Signature Isolation Forest? 

How we Achieved 11% Growth by Leveraging Personalization and Segment-based Product Flows

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring the Gradient of a Scalar in Dark Matter Velocity Analysis

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar C Covariance Matrix To treat the total likelihood as the product of the individual likelihoods in each frequency bin, we must check that the covariance matrix is diagonal. We will consider a signal-only analysis, discarding the noise, since the noise merely adds to the power and is uncorrelated between different frequency bins. We may write the values of the three peaks as We wish to compute the quantity We can do this using the expression for the raw moments, where, for us, σ = 1/ √ 2. Aside from this, we need to know that We then get that Crucially, we get that the covariance between peaks is 0, allowing us to treat them as statistically independent and hence permitting us to express the total likelihood as the product of the individual likelihoods. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar C Covariance Matrix To treat the total likelihood as the product of the individual likelihoods in each frequency bin, we must check that the covariance matrix is diagonal. We will consider a signal-only analysis, discarding the noise, since the noise merely adds to the power and is uncorrelated between different frequency bins. We may write the values of the three peaks as We wish to compute the quantity We can do this using the expression for the raw moments, where, for us, σ = 1/ √ 2. Aside from this, we need to know that We then get that Crucially, we get that the covariance between peaks is 0, allowing us to treat them as statistically independent and hence permitting us to express the total likelihood as the product of the individual likelihoods. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

nalyzing Covariance Matrix for Independent Likelihoods in Frequency Bins

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar A Equipartition between Longitudinal and Transverse Modes The equation of motion for the field Ψ (written in Fourier space) is Now, we wish to decompose the field into transverse and longitudinal pieces. Let us call them ηk and ζk. That is, where Notice that not only are the longitudinal and transverse pieces coupled to one another, but more importantly they source each other. Hence, we expect virialization between them due to such non-linear gravitational dynamics during DM halo formation, ultimately leading to equipartition of power amongst them. That is, 2/3 of the total power would be in the transverse sector, with the remaining 1/3 in the longitudinal sector. Using codes based on the split-Fourier technique [73, 91–93], we have performed simulations of the Schrodinger -Poisson vector system [31, 64, 94] (physical space version of Eq. (A.1)): This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar A Equipartition between Longitudinal and Transverse Modes The equation of motion for the field Ψ (written in Fourier space) is Now, we wish to decompose the field into transverse and longitudinal pieces. Let us call them ηk and ζk. That is, where Notice that not only are the longitudinal and transverse pieces coupled to one another, but more importantly they source each other. Hence, we expect virialization between them due to such non-linear gravitational dynamics during DM halo formation, ultimately leading to equipartition of power amongst them. That is, 2/3 of the total power would be in the transverse sector, with the remaining 1/3 in the longitudinal sector. Using codes based on the split-Fourier technique [73, 91–93], we have performed simulations of the Schrodinger -Poisson vector system [31, 64, 94] (physical space version of Eq. (A.1)): This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Equipartition Between Longitudinal and Transverse Modes

Advancing the Detection of Ultralight Vector Dark Matter through Novel Analytical Strategies

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar 5 Future Directions 5.1 Longer Observation Times 5.2 Expanding the Mass Window This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar 5 Future Directions 5.1 Longer Observation Times 5.2 Expanding the Mass Window This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Longer Observation Times and Expanding the Mass Window

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar 4.1 Recasting Generalised Limits onto B − L Dark Matter To recast our limits on β shown in Fig. 4 into one on the parameter of interest for this model, the gauge coupling strength gB−L, we must define four quantities. Firstly, we must make clear what the quantity A is for this experiment, which will depend on both the model and the signal of interest for this sensor. Secondly, we must choose an observation time, Tobs. Thirdly, we must elaborate on what the concrete noise profile, σ(ω), for this type of experiment is. Lastly, and least importantly following from our discussion above, we must choose a latitude for the experiment. Once these are known, Eq. (2.8) can be re-arranged for gB−L, giving us our model- and experiment-specific 90% CL limit. For B − L coupled dark photon dark matter, the relevant signal is a differential acceleration. This is given by Eq. (2.6) in the time domain, with a now concrete choice for A, Here, gB−L is the gauge coupling strength of the model, ∆ij is the differential B −L charge per nucleon between materials i and j, given by For our observation time, we take Tobs = 10T⊕ to be firmly in the regime where the three peaks can be resolved. For light cavities, which may only be able to remain coherent over the scale of hours rather than days, such a runtime is optimistic. However, since our aim here is merely to showcase how our method can be used concretely and compare to the works of Refs. [36, 38], we do not see this as an issue. Our strategy is general and can be applied to any axial sensor and vector-like DM model, and we have settled on this configuration only for the sake of argument; other sensor technologies, such a magnetically levitated sensors, do not have this issue. Moreover, multiple cavities or data-stacking techniques can be employed to mitigate this issue. Our choices for all of the above parameters except for the laser power, which we expand on below, are summarised in Table 1. These are in keeping with the choices made in Refs. [36, 38]. The choice of where to tune the laser power is critical to give us competitive limits for a wide range of dark matter masses. We have found that we can achieve excellent limits for low dark matter masses, which are of most relevance to our work, by tuning the laser power such that the back-action and shot-noise components are minimised at low We note that this choice of power tuning is different from the strategies usually employed in other studies. Typically, the laser power is tuned so that the measurement-added noise is either minimised on resonance or, as LIGO implements it, well above resonance [88]. However, we found that both of these choices detriments the limit that we can draw at low masses, increasing the background at low frequencies beyond the thermal noise floor. For these other strategies to be beneficial at both the respective frequency targets and at low frequencies, the thermal noise would have to be significantly lower than that achieved with our choice of sensor mass, quality factor, and bath temperature. In the treatment of our backgrounds, we have neglected seismic noise, which can become important at frequencies below ∼ 10 Hz. Since we require the use of two materials to measure the differential acceleration peculiar to B − L DM, we can envision constructing two sensors: one with a moveable mirror made of material i and another of material j. By subtracting their signals, we both isolate the differential acceleration signature and remove backgrounds common to both—this includes the seismic noise component [63] We show our limits in Fig. 5 for three choices of resonance frequency: 1 Hz, 10 Hz and 100 Hz. For all but the last of these frequencies, we are able to exclude new regions of the UDLM B − L parameter space, best excluded by the fifth-force satellite experiment MICROSCOPE [89] and torsion-balance experiment E¨ot-Wash [90]. This showcases the power of such sensors in searching for this DM candidate, as was first pointed out in Ref. [63]. In a single-peak analysis focusing solely on the Compton peak, our limits would be weakened by approximately a factor of 2, as can be seen from Fig. 4. This difference becomes more dramatic for sensors located closer to the equator. We note that σ(ω), though mostly slowly varying in the frequency width ∆ω = ω⊕, does exhibit a large gradient around the resonance frequency of the cavity. In the neighbourhood of this frequency, our assumption that σ(ω) does not vary greatly in the above range is incorrect, and a more careful analysis in which all three peaks take on different noise levels would have to be conducted for more representative limits. However, we do not expect that our limits would differ greatly from our calculation and, at any rate, would still smoothly join to the regimes on either side of the resonance frequency, where our assumption holds. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar 4.1 Recasting Generalised Limits onto B − L Dark Matter To recast our limits on β shown in Fig. 4 into one on the parameter of interest for this model, the gauge coupling strength gB−L, we must define four quantities. Firstly, we must make clear what the quantity A is for this experiment, which will depend on both the model and the signal of interest for this sensor. Secondly, we must choose an observation time, Tobs. Thirdly, we must elaborate on what the concrete noise profile, σ(ω), for this type of experiment is. Lastly, and least importantly following from our discussion above, we must choose a latitude for the experiment. Once these are known, Eq. (2.8) can be re-arranged for gB−L, giving us our model- and experiment-specific 90% CL limit. For B − L coupled dark photon dark matter, the relevant signal is a differential acceleration. This is given by Eq. (2.6) in the time domain, with a now concrete choice for A, Here, gB−L is the gauge coupling strength of the model, ∆ij is the differential B −L charge per nucleon between materials i and j, given by For our observation time, we take Tobs = 10T⊕ to be firmly in the regime where the three peaks can be resolved. For light cavities, which may only be able to remain coherent over the scale of hours rather than days, such a runtime is optimistic. However, since our aim here is merely to showcase how our method can be used concretely and compare to the works of Refs. [36, 38], we do not see this as an issue. Our strategy is general and can be applied to any axial sensor and vector-like DM model, and we have settled on this configuration only for the sake of argument; other sensor technologies, such a magnetically levitated sensors, do not have this issue. Moreover, multiple cavities or data-stacking techniques can be employed to mitigate this issue. Our choices for all of the above parameters except for the laser power, which we expand on below, are summarised in Table 1. These are in keeping with the choices made in Refs. [36, 38]. The choice of where to tune the laser power is critical to give us competitive limits for a wide range of dark matter masses. We have found that we can achieve excellent limits for low dark matter masses, which are of most relevance to our work, by tuning the laser power such that the back-action and shot-noise components are minimised at low We note that this choice of power tuning is different from the strategies usually employed in other studies. Typically, the laser power is tuned so that the measurement-added noise is either minimised on resonance or, as LIGO implements it, well above resonance [88]. However, we found that both of these choices detriments the limit that we can draw at low masses, increasing the background at low frequencies beyond the thermal noise floor. For these other strategies to be beneficial at both the respective frequency targets and at low frequencies, the thermal noise would have to be significantly lower than that achieved with our choice of sensor mass, quality factor, and bath temperature. In the treatment of our backgrounds, we have neglected seismic noise, which can become important at frequencies below ∼ 10 Hz. Since we require the use of two materials to measure the differential acceleration peculiar to B − L DM, we can envision constructing two sensors: one with a moveable mirror made of material i and another of material j. By subtracting their signals, we both isolate the differential acceleration signature and remove backgrounds common to both—this includes the seismic noise component [63] We show our limits in Fig. 5 for three choices of resonance frequency: 1 Hz, 10 Hz and 100 Hz. For all but the last of these frequencies, we are able to exclude new regions of the UDLM B − L parameter space, best excluded by the fifth-force satellite experiment MICROSCOPE [89] and torsion-balance experiment E¨ot-Wash [90]. This showcases the power of such sensors in searching for this DM candidate, as was first pointed out in Ref. [63]. In a single-peak analysis focusing solely on the Compton peak, our limits would be weakened by approximately a factor of 2, as can be seen from Fig. 4. This difference becomes more dramatic for sensors located closer to the equator. We note that σ(ω), though mostly slowly varying in the frequency width ∆ω = ω⊕, does exhibit a large gradient around the resonance frequency of the cavity. In the neighbourhood of this frequency, our assumption that σ(ω) does not vary greatly in the above range is incorrect, and a more careful analysis in which all three peaks take on different noise levels would have to be conducted for more representative limits. However, we do not expect that our limits would differ greatly from our calculation and, at any rate, would still smoothly join to the regimes on either side of the resonance frequency, where our assumption holds. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Stories, News & Mentions about AMPLITUDE

Amplitude’s Stories on HackerNoon

Amplitude's News & Mentions